Integral over head-light contour via Residue Theorem

This is a challenge question that I already know how to solve: Let $\displaystyle f(z)=\sqrt{1-z^2}$ and consider a smooth (differentiable) map $\displaystyle f(H(t))$ where $\displaystyle H$ is the general shape shown in the first plot below (loops around the branch-points,then looping around to close the contour). That is, the contour traverses an analytic path over multiple branches of the integrand. Show via the Residue Theorem: